Chapter 2 Hardy Inequalities on Homogeneous Groups

This chapter is devoted to Hardy inequalities and the analysis of their remainders in different forms. Moreover, we discuss several related inequalities such as Rellich inequalities and uncertainty principles.


Hardy inequalities and sharp remainders
In this section we analyse the anisotropic version of the classical L p -Hardy inequality where ∇ is the standard gradient in R n , |x| E = x 2 1 + · · · + x 2 n is the Euclidean norm, f ∈ C ∞ 0 (R n ), and the constant p n−p is known to be sharp. We also discuss in detail its critical cases and remainder estimates. As consequences, we derive Rellich type inequalities and the corresponding uncertainty principles.

Hardy inequality and uncertainty principle
First we establish the L p -Hardy inequality and derive a formula for the remainder on a homogeneous group G of homogeneous dimension Q ≥ 2. The radial operator R from (1.30) is entering the appearing expressions.
Theorem 2.1.1 (Hardy inequalities on homogeneous groups). Let | · | be any homogeneous quasi-norm on G.
(i) Let f ∈ C ∞ 0 (G\{0}) be a complex-valued function. Then we have where the constant p Q−p is sharp. Moreover, the equality in (2.2) is attained if and only if f = 0.
1. In the case of G = R n and |x| = |x| E = x 2 1 + · · · + x 2 n the Euclidean norm, we have Q = n and R = ∂ r is the usual radial derivative, and (2.2) implies the classical Hardy inequality (2.1). Indeed, in this case for 1 < p < n inequality (2.2) yields in view of the Cauchy-Schwarz inequality for the Euclidean norm. An interesting feature of the Hardy inequality in Part (i) is that the constant in (2.2) is sharp for any homogeneous quasi-norm | · |. 2. In the setting of Part 1 above the remainder formula (2.3) for the Euclidean norm | · | E in R n was analysed by Ioku, Ishiwata and Ozawa [IIO17].
3. In Theorem 2.1.1, Part (ii) implies Part (i). To show it one can notice that the right-hand side of (2.3) is non-negative, which implies that for any real-valued f ∈ C ∞ 0 (G\{0}). Moreover, by using the following identity we obtain the same inequality for all complex-valued functions: for all z ∈ C we have |Re(z) cos θ + Im(z) sin θ| p dθ, (2.8) which is a consequence of the decomposition of a complex number z = r(cos φ + i sin φ).
That is, we obtain inequality (2.2), and also that the constant p Q−p is sharp, in view of the remainder formula. Now let us show that this constant is attained only for f = 0. Identity (2.8) says that it is sufficient to look only for real-valued functions f . If the right-hand side of (2.3) vanishes, then we must have u = v, that is, This also means that Ef = − Q−p p f . Lemma 1.3.1 implies that f is positively homogeneous of order − Q−p p , i.e., there exists a function h : ℘ → C such that where ℘ is the unit sphere for the quasi-norm | · |. It confirms that f cannot be compactly supported unless it is identically zero.
4. The identity (2.8) has been often used in similar estimates for passing from real-valued to complex-valued functions, see, e.g., Davies [Dav80,p. 176]. 5. Let us denote by H 1 R (G) the functional space of the functions f ∈ L 2 (G) with Rf ∈ L 2 (G). Then Theorem 2.1.1 can be extended for functions in H 1 R (G), that is, the proof of (2.2) given above works in this case. As for the sharpness and the equality in (2.2), having (2.9) also implies that f (x) is not in L p (G) unless h = 0 and f = 0.
Remark 2.1.2, Part 2, shows that (2.3) implies Part (i) of Theorem 2.1.1, that is, we only need to prove Parts (ii) and (iii). However, we now give an independent proof of (2.2) for complex-valued functions without relying on the formula (2.8). We see that this argument will be also useful in the proof of Part (ii).
Proof of Theorem 2.1.1. Proof of Part (i). Using the polar decomposition from Proposition 1.2.10, a direct calculation shows that (2.10) Now by the Hölder inequality with 1 p + 1 q = 1 we obtain This proves inequality (2.2) in Part (i).
Proof of Part (ii). Since For a real-valued f the formula (2.10) becomes and (2.11) becomes To show the last equality in (2.13), we observe the identity, for real numbers u = v, using the integral expression for the remainder in the Taylor expansion formula. Combining (2.13) with (2.12) we arrive at It completes the proof of Part (ii).
Then we have that is, (2.5) is proved.
As a direct consequence of the inequality (2.2) we obtain the corresponding uncertainty principle: Corollary 2.1.3 (Uncertainty principle on homogeneous groups). For every com- (2.14) Here Q ≥ 2, | · | is an arbitrary homogeneous quasi-norm on G, 1 < p < Q and 1 p + 1 q = 1.
Proof. The inequality (2.7) and the Hölder inequality imply that This shows (2.14).
Remark 2.1.4. In the Abelian case G = (R n , +) with the standard Euclidean distance |x| E , we have Q = n, so that (2.14) with p = q = 2 and n ≥ 3 implies the uncertainty principle which in turn implies the classical uncertainty principle for G ≡ R n :

Weighted Hardy inequalities
In this section G is a homogeneous group of homogeneous dimension Q ≥ 3. Let | · | be an arbitrary homogeneous quasi-norm on G. Here, we are going to discuss weighted Hardy inequalities on G which are the consequences of exact equalities.
Theorem 2.1.5 (Weighted Hardy identity in L 2 (G)). Let G be a homogeneous group of homogeneous dimension Q ≥ 3 and let | · | be a homogeneous quasi-norm on G.
Then for every complex-valued function f ∈ C ∞ 0 (G\{0}) and for any α ∈ R we have the equality . (2.16) The equality (2.16) implies many different inequalities. For instance, by taking α = 1 and simplifying its coefficient, for any Q ≥ 3 we obtain the identity . (2.17) By dropping the last term in (2.16) which is non-negative we obtain: Corollary 2.1.6 (Weighted Hardy inequality in L 2 (G)). Let Q ≥ 3 and let α ∈ R be such that . (2.18) Here the constant 2 |Q−2−2α| is sharp and it is attained if and only if f = 0. Remark 2.1.7.
1. It is interesting to note that the constant 2 |Q−2−2α| in (2.18) is sharp for any homogeneous quasi-norm | · | on G.
In the case of the Euclidean distance |x| E = x 2 1 + · · · + x 2 n , by the Cauchy-Schwarz inequality we obtain the following estimate: for all α ∈ R and for any f ∈ C ∞ 0 (R n \{0}).

Hardy inequalities with homogeneous weights have been also considered by
Hoffmann-Ostenhof and Laptev [HOL15]. There are also further many-particle versions of such inequalities, see [HOHOLT08] and many further references therein. We will discuss some of such inequalities in Section 6.11 and Section 6.12.
6. Theorem 2.1.5 was established in [RS17b]. Its extension from L 2 to L p spaces presented in Theorem 2.1.8 was made in [Ngu17].
Proof of Theorem 2.1.5. We first observe the equality for any α ∈ R, which follows from and hence, by using (1.30), we have Then using (2.21) we can write By applying (2.5) to the function f |x| α and using (2.21) we have that .
In addition, a direct calculation using the polar decomposition in Proposition 1.2.10 shows that In conclusion, combining these identities we arrive at , yielding (2.16).
To present a weighted L p -Hardy inequality on G we will use the following function R p in analogy to I p in (2.4). For ξ, η ∈ C we denote By the convexity of the function z → |z| p we see that R p (ξ, η) ≥ 0 is non-negative and R p (ξ, η) = 0 and if and only if ξ = η. If ξ, η ∈ R, we then have analogous to I p in (2.4).
Theorem 2.1.8 (Weighted Hardy identity in L p (G)). Let G be a homogeneous group of homogeneous dimension Q. Let 1 < p < Q and α ∈ R. Then for any homogeneous quasi-norm | · | on G and for all complex (2.23) By dropping the last term in (2.23) which is non-negative we obtain: Corollary 2.1.9 (Weighted Hardy inequality in L p (G)). Let 1 < p < Q and let α ∈ R. Then for all complex-valued .
(2.24) Proof of Theorem 2.1.8. We can assume that Q − p(1 + α) = 0, otherwise there is nothing to prove. A direct calculation gives which implies the identity (2.23).
Proof of Corollary 2.1.9. The equality (2.23) implies inequality (2.24) since the last term in (2.23) is non-negative. Let us now show the sharpness of the constant. For this we approximate the function r −(Q−p(1+α))/p by smooth compactly supported functions, for details of such an argument see also the proof of Theorem 3.1.4. Using (2.23) it follows that the equality in (2.24) holds if and only if or, equivalently, In turn, this is equivalent to By Proposition 1.3.1 it follows that f is positively homogeneous of order −(Q − p(1 + α))/p. Since |f |/|x| 1+α is in L p (G), it follows that f = 0.

Hardy inequalities with super weights
In this section we discuss sharp L p -Hardy type inequalities with super weights, i.e., with weights of the form (a + b|x| α ) β p |x| m . (2.25) Such weights are sometimes called the super weights because of the arbitrariness of the choice of any homogeneous quasi-norm as well as a wide range of parameters. However, all the inequalities can be obtained with best constants.
Theorem 2.1.10 (Hardy inequalities with super weights). Let G be a homogeneous group of homogeneous dimension Q ≥ 1. Let a, b > 0 and 1 < p < ∞. Then we have the following inequalities: . (2.26) . (2.27) Proof of Theorem 2.1.10. Proof of Part (i). We can assume Q = pm + p since in the case Q = pm + p there is nothing to prove. As usual with (r, y) = (|x|, x |x| ) ∈ (0, ∞) × ℘ on G, where ℘ := {x ∈ G : |x| = 1}, using the polar decomposition in Proposition 1.2.10 and integrating by parts, we obtain Now by using Hölder's inequality we arrive at the inequality which gives (2.26). We need to check the equality condition in the above Hölder inequality in order to show the sharpness of the constant. Setting where C ∈ R, C = 0 and Q = pm + p a direct calculation shows which satisfies the equality condition of the Hölder inequality. This gives the sharpness of the constant Q−pm−p p in the inequality (2.26).
Proof of Part (ii). Here we can assume that Q = pm + p − αβ since for Q = pm+p−αβ there is nothing to prove. Using the polar decomposition in Proposition 1.2.10, as before we have the equality (2.28). Since αβ < 0 and pm − αβ < Q − p we obtain By Hölder's inequality, it follows that To show the sharpness of the constant we will check the equality condition in the above Hölder inequality. Thus, by taking , which satisfies the equality condition in Hölder's inequality. This gives the sharpness of the constant Q−pm−p+αβ p in (2.27).

Hardy inequalities of higher order with super weights
The iteration process gives the following higher-order L p -Hardy type inequalities with super weights. Here as before, G is a homogeneous group of homogeneous dimension Q ≥ 1 and | · | is a homogeneous quasi-norm on G.
2. In the Euclidean case G = R n and | · | = | · | E the Euclidean norm, the super weights in the form (2.25) have appeared in [GM08], together with some applications to problems for differential equations. The case of homogeneous groups, as well as the iterative higher order estimates as in Theorem 2.1.11 were analysed in [RSY17b, RSY18b].
Combining this with (2.32) we obtain .

This iteration process gives
, completing the proof.

Two-weight Hardy inequalities
In this section, using the method of factorization of differential expressions, we obtain Hardy type inequalities with two general weights φ(x) and ψ(x). The idea of the factorization method can be best illustrated by the following example of an estimate due to Gesztesy and Littlejohn [GL17].
Example 2.1.13 (Gesztesy and Littlejohn two-parameter inequality). Let α, β ∈ R, x ∈ R n \{0} and n ≥ 2. Let us define the operator One readily checks that its formal adjoint is given by Using the non-negativity of the operator (2.33) By choosing particular values of α and β it can be checked that this inequality yields classical Rellich and Hardy-Rellich type inequalities as special cases, see [GL17] for details. On the other hand, using the non-negativity of the operator T α,β T + α,β , it was shown in [RY17] that for α, β ∈ R and n ≥ 2, and for all f ∈ C ∞ 0 (R n \{0}) we can deduce another two-parameter inequality The following result is a two-weight inequality on general homogeneous groups with general weights.
Theorem 2.1.14 (Two-weight Hardy inequality). Let G be a homogeneous group of homogeneous dimension Q ≥ 3 and let |·| be a homogeneous quasi-norm on G. Let φ, ψ ∈ L 2 loc (G\{0}) be any real-valued functions such that Rφ, Rψ ∈ L 2 loc (G\{0}). 1. If we take φ(x) ≡ 1 in Theorem 2.1.14, we obtain for all α ∈ R and for all In the case when we take b = a + 1, we get Then, by maximizing the constant (α(Q − 2a − 2) − α 2 ) with respect to α we obtain the weighted Hardy inequality from Corollary 2.1.6, namely, for which it is known that the constant in (2.37) is sharp.
After maximizing the above constant with respect to α we obtain the critical Hardy inequality Proof of Theorem 2.1.14. Let us introduce the one-parameter differential expression One can readily calculate for the formal adjoint operator of T α on C ∞ 0 (G\{0}): Thus, the formal adjoint operator of T α has the form where ℘ is the quasi-sphere as in (1.12), and using Proposition 1.2.10 one calculates (2.39) where we set Now we simplify the sum of the terms I 1 , I 2 , I 3 , I 5 and I 6 . By a direct calculation we obtain Now we calculate I 2 as follows, For I 3 , one has For I 5 , we have Finally, for I 6 we obtain Putting (2.40)-(2.41) in (2.39), we obtain that which implies (2.35). Thus, we have obtained (2.35) using the non-negativity of T + α T α . Now we can obtain (2.36) using the non-negativity of T α T + α . Similar to the above we calculate Using the non-negativity of T α T + α we get where Taking into account these and (2.40)-(2.41), we obtain (2.36).
To finish this section, we observe another version of weighted Hardy inequality with the radial derivative. Anticipating the material presented in later chapters, the proof will be based on the integral Hardy inequality from Theorem 5.1.1.
Theorem 2.1.16 (Weighted Hardy inequality for radial functions). Let G be a homogeneous group of homogeneous dimension Q. Let φ > 0, ψ > 0 be positive weight functions on G and let 1 < p ≤ q < ∞. Then there exists a positive constant Consequently, usingf (0) = 0, we have

Critical Hardy inequalities
In this section we discuss critical Hardy inequalities. The critical behaviour may be manifested with respect to different parameters. For example, one major critical case arises when we have p = Q in (2.2). In this case, in the Euclidean case of R n with p = Q = n it is known that the Hardy inequality (2.1) fails for any constant, see, e.g., [ET99] and [IIO16a], and references therein. In such critical cases it is natural to expect the appearance of the logarithmic terms.
One version of such a critical case is the inequality In fact, this inequality is a special case (with p = Q) of the following more general family of critical inequalities derived in Theorem 2.2.1, namely, (2.45) for all 1 < p < ∞. Here the constant p p−1 in (2.45) is sharp but is in general unattainable.
The inequalities (2.45) are all critical with respect to their weight |x| − Q p in L p -space because its pth power gives |x| −Q which is the critical order for the integrability at zero and at infinity in L p (G).
Moreover, we show another type of a critical Hardy inequality on general homogeneous groups with the logarithm being on the right-hand side: (2.46) Furthermore, we also give improved versions of the Hardy inequality (2.1) on quasi-balls of homogeneous (Lie) groups, the so-called Hardy-Sobolev type inequalities.

Critical Hardy inequalities
First we discuss a family of generalized critical Hardy inequalities on the homogeneous group G mentioned in (2.45). In this section, G is a homogeneous group of homogeneous dimension Q ≥ 2 and | · | is an arbitrary homogeneous quasi-norm on G.
where the constant p p−1 is sharp. Moreover, denoting for each R > 0 we have the following expression for the remainder:

48)
where I is defined by Remark 2.2.2.

For p = Q the inequality (2.47) becomes
where σ is the Borel measure on ℘ and the contribution on the boundary at r = R vanishes due to the inequalities By using the formula (1.30) we obtain Similarly, one has

This implies that
and I is defined by Thus, we establish This proves the equality (2.48) and the inequality (2.47) since the last term is non-positive.
We have the following consequence of Theorem 2.2.1: Corollary 2.2.3 (Critical uncertainty type principles). Let 1 < p < ∞ and f ∈ C ∞ 0 (G\{0}). Then for any R > 0 and 1 and also The formula (2.52) is proved. The proof of (2.53) is similar so we can omit it.

Another type of critical Hardy inequality
In this section we analyse another type of a critical Hardy inequality when the logarithmic term appears on the other side of the inequality. This extends inequality (2.38) that was obtained for p = 2 by a factorization method.
Theorem 2.2.4 (Critical Hardy inequality). Let G be a homogeneous group of homogeneous dimension Q ≥ 1 and let | · | be a homogeneous quasi-norm on G. Let 1 < p < ∞. Then for any complex-valued function (2.55) 2. In the Euclidean case G = (R n , +) we have Q = n, so for any quasi-norm | · | on R n the inequality (2.54) implies a new inequality with the optimal constant: . (2.56) If we take now the standard Euclidean distance |x| E = x 2 1 + · · · + x 2 n , it follows that we have where ∇ is the standard gradient in R n . The constants in the above inequalities are sharp. 3. The inequality (2.57) and its consequence are analogous to the critical Hardy inequality of Edmunds and Triebel [ET99] that they showed in R n for the Euclidean norm | · | E in bounded domains B ⊂ R n : with sharp constant n n−1 , which was also discussed in [AS06]. This inequality was also shown to be equivalent to the critical case of the Sobolev-Lorentz inequality. However, a different feature of (2.57) compared to (2.58) is that the logarithmic term enters the other side of the inequality. Inequalities of this type have been investigated in [RS16a].
Proof of Theorem 2.2.4. Let R > 0 be such that suppf ⊂ B(0, R). A direct calculation using the polar decomposition in Proposition 1.2.10 with integration by parts yields and by using the Hölder inequality, we obtain that which gives (2.54). Now it remains to show the optimality of the constant, so we need to check the equality condition in the above Hölder inequality. Let us consider the test function h(x) = log |x|.
Thus, we have which satisfies the equality condition in Hölder's inequality. This gives the optimality of the constant p p in (2.54).
As usual, the Hardy inequality implies the corresponding uncertainty principle: Corollary 2.2.6 (Another type of critical uncertainty principle). Let G be a homogeneous group of homogeneous dimension Q ≥ 2. Let | · | be a homogeneous quasi-norm on G. Then for each f ∈ C ∞ 0 (G\{0}) we have Proof. From the inequality (2.54) we get where we have used the Hölder inequality in the last line. This shows (2.59).

Critical Hardy inequalities of logarithmic type
In this section we present yet another logarithmic type of critical Hardy inequalities on the homogeneous group G of homogeneous dimension Q ≥ 1. As usual, let | · | be a homogeneous quasi-norm on G.
Proof of Theorem 2.2.7. For a quasi-ball B(0, R) we have using the polar decomposition in Proposition 1.2.10 that where p−γ+1 > 0, so that the boundary term at r = R vanishes due to inequalities Then by the Hölder inequality we get The inequalities (2.61) and (2.62) imply (2.60). Furthermore, the optimality of the constant in (2.60) is proved exactly in the same way as in the Euclidean case (see [MOW15b, Section 3]).
Proof of Corollary 2.2.8. By (2.60), we have and using Hölder's inequality, we obtain .
Remark 2.2.9. When γ = p, the statement in Theorem 2.2.7 appeared in [RS16a, Theorem 3.1] in the form

Remainder estimates
In this section we analyse the remainder estimates for L p -weighted Hardy inequalities with sharp constants on homogeneous groups. In addition, other refined versions involving a distance and the critical case p = Q = 2 on the quasi-ball are discussed.
The analysis of remainder terms in Hardy inequalities has a long history initiated by Brézis and Nirenberg in [BN83], with subsequent works by Brézis and Lieb [BL85] for Hardy-Sobolev inequalities, Brézis and Vázquez in [BV97, Section 4]. Nowadays there is a lot of literature on this subject and this section will contain some further references on this subject.

Remainder estimates for L p -weighted Hardy inequalities
Let G be a homogeneous group of homogeneous dimension Q ≥ 3 and let | · | be a homogeneous quasi-norm on G. We now present a family of remainder estimates for the weighted L p -Hardy inequalities, with a freedom of choosing the parameter b ∈ R.
Theorem 2.3.1 (Remainder estimates for L p -weighted Hardy inequalities). Let and let Due to the positivity of the last remainder term, Theorem 2.3.1 implies the L p -weighted Hardy inequalities with the radial derivative. In the case of the Euclidean norm, they reduce to the usual L p -weighted Hardy estimates: Remark 2.3.2 (L p -weighted Hardy inequalities).
1. If we take b = Q(p−1) p we have C p = 0, and then inequality (2.65) gives the L p -weighted Hardy inequalities with sharp constant on G: for all complex-valued functions f ∈ C ∞ 0 (G\{0}). Such inequalities on homogeneous groups have been investigated in [RS17b], and their remainders have been analysed in [RSY18b]. 2. If G = (R n , +) with Q = n, the inequality (2.67) gives the L p -weighted Hardy inequalities with sharp constant for any quasi-norm on R n : For any where −∞ < α < n−p p and 2 ≤ p < n, and where ∇ is the standard gradient in R n . Now, if we take the Euclidean norm |x| E = x 2 1 + x 2 2 + · · · + x 2 n , by using the Schwarz inequality we obtain the Euclidean form of the L p -weighted Hardy inequalities with sharp constants: for any complex-valued function f ∈ C ∞ 0 (R n \{0}). 3. Moreover, for any function f ∈ C ∞ 0 (R n \{0}) and for any b ∈ R, we have where ∇ is the standard gradient in R n . As in Part 2 above, by the Schwarz inequality with the usual Euclidean distance | · | E , we obtain for all complex-valued functions f ∈ C ∞ 0 (R n \{0}) and for any b ∈ R, where the constant C p is sharp. 4. In R n a variant of inequality (2.65) is known for radially symmetric functions.
Let n ≥ 3, 2 ≤ p < n and −∞ < α < n−p p . Let N ∈ N, t ∈ (0, 1), . Then there exists a constant C > 0 such that the inequality holds for all radially symmetric functions f ∈ W 1,p 0,α (R n ), f = 0, where W 1,p 0,α (R n ) is an appropriate Sobolev type space. For α = 0 this was shown by Sano and Takahashi [ST17] and then extended in [ST18a] for any −∞ < α < n−p p . 5. The constant c p in (2.66) appears in view of the following result that will be also of use in the determination of this constant: 6. Remainder estimates of different forms are possible. In general, it is known from Ghoussoub and Moradifam [GM08] that there are no strictly positive functions V ∈ C 1 (0, ∞) such that the inequality holds for all Sobolev space functions f ∈ W 1,2 (R n ). At the same time, Cianchi and Ferone showed in [CF08] that for all 1 < p < n there exists a constant holds for all real-valued weakly differentiable functions f in R n such that f and |∇f | ∈ L p (R n ) go to zero at infinity, where with p * = np n−p , and L τ,σ (R n ) is the Lorentz space for 0 < τ ≤ ∞ and 1 ≤ σ ≤ ∞. In the case of a bounded domain Ω, Wang and Willem [WW03] for p = 2 and Abdellaoui, Colorado and Peral [ACP05] for 1 < p < ∞ investigated other expressions of remainders, see also [ST17] and [ST18a] for more details.
In the following proof we will rely on a useful feature that some estimates involving radial derivatives of the Euler operator can be proved first for radial functions, and then extended to non-radial ones by a more abstract argument, see Section 1.3.3.
In view of (2.74), we obtain the equality for any θ ∈ R. Then, it is easy to see that (2.75) and (2.76) imply that (2.65) holds also for all non-radial functions.

Critical and subcritical Hardy inequalities
Here we discuss the relation between the critical and the subcritical Hardy inequalities on homogeneous groups. We formulate this relation for functions that are radially symmetric with respect to a homogeneous quasi-norm | · | on G.
Proposition 2.3.4 (Critical and subcritical Hardy inequalities). Let G be a homogeneous group of homogeneous dimension Q ≥ 3 and let G be a homogeneous group of homogeneous dimension m ≥ 2, and assume that Q ≥ m + 1. Let | · | denote homogeneous quasi-norms on G and on G. Then for any non-negative radially symmetric function g ∈ C 1 0 (B m (0, R)\{0}), there exists a non-negative radially symmetric function f ∈ C 1 0 (B Q (0, 1)\{0}) such that Proof of Proposition 2.3.4. Let r = |x|, x ∈ G and s = |z|, z ∈ G, where G is a homogeneous group of homogeneous dimension m. Let us define a radial function Here we see that s (r) > 0 for r ∈ [0, 1] and s(0) = 0, s(1) = R. Since g(s) ≡ 0 near s = R, we also note that f ≡ 0 near r = 1.
Then a direct calculation shows yielding (2.77).

A family of Hardy-Sobolev type inequalities on quasi-balls
Let G be a homogeneous group of homogeneous dimension Q ≥ 3. It will be convenient to denote the dilations by δ r (x) = rx in the following formulations. Here we discuss another type of Hardy-Sobolev inequalities for functions supported in balls of radius R. As usual, we denote by B(0, R) a quasi-ball of radius R around 0 with respect to the quasi-norm | · |.
1. Theorem 2.3.5 could have been formulated for functions f ∈ C ∞ 0 (G\{0}) choosing R > 0 such that suppf ⊂ B(0, R). The introduction of R into the notation is essential here since the dilated function f δR(x) |x| appears in the inequality (2.78). 2. In the Euclidean setting with the Euclidean norm inequalities in Theorem 2.3.5 have been studied in [MOW13b].

Improved Hardy inequalities on quasi-balls
For p = Q = 2 and any homogeneous quasi-norm | · | on G we have the following refinement of Theorem 2.3.7 with an estimate for a remainder.

90)
where |℘| is the measure of the unit quasi-sphere in G and where α = α(q, L) = Q−1 Q q + L + 2. The proof is complete. (2.100) According to Remark 2.2.9 for v ∈ C ∞ 0 (G\{0}) with p = Q and (2.100), it follows that Thus, we arrive at for all T > 0. This completes the proof of Theorem 2.4.4.
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